Comptes Rendus
no. 5
Partial differential equations
An example of non-decreasing solution for the KdV equation posed on a bounded interval
[Un exemple de solution non décroissante de l'équation de KdV posée sur un intervalle borné]

Présenté par : the Editorial Board

Gleb Germanovitch Doronin1 ; Fábio M. Natali1
1 Departamento de Matemática, Universidade Estadual de Maringá, 87020-900, Maringá, PR, Brazil
Comptes Rendus. Mathématique, Volume 352 (2014) no. 5, pp. 421-424.

On considère un problème avec donnée initiale et au bord pour l'équation de KdV posée sur un intervalle borné. La théorie des fonctions elliptiques de Jacobi est utilisée pour obtenir un nouveau type d'ondes stationnaires qui sont périodiques en espace avec une période égale à une longueur d'intervalle. Les propriétés de ces solutions sont étudiées.

An initial-boundary value problem for the KdV equation posed on a bounded interval is considered. The theory of Jacobi elliptic functions is used to obtain a new kind of stationary waves which are spatially periodic with a period equal to an interval length. The properties of those solutions are studied.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.02.001
Gleb Germanovitch Doronin 1 ; Fábio M. Natali 1

1 Departamento de Matemática, Universidade Estadual de Maringá, 87020-900, Maringá, PR, Brazil 
@article{CRMATH_2014__352_5_421_0,
     author = {Gleb Germanovitch Doronin and F\'abio M. Natali},
     title = {An example of non-decreasing solution for the {KdV} equation posed on a bounded interval},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {421--424},
     publisher = {Elsevier},
     volume = {352},
     number = {5},
     year = {2014},
     doi = {10.1016/j.crma.2014.02.001},
     language = {en},
}
TY  - JOUR
AU  - Gleb Germanovitch Doronin
AU  - Fábio M. Natali
TI  - An example of non-decreasing solution for the KdV equation posed on a bounded interval
JO  - Comptes Rendus. Mathématique
PY  - 2014
SP  - 421
EP  - 424
VL  - 352
IS  - 5
PB  - Elsevier
DO  - 10.1016/j.crma.2014.02.001
LA  - en
ID  - CRMATH_2014__352_5_421_0
ER  - 
%0 Journal Article
%A Gleb Germanovitch Doronin
%A Fábio M. Natali
%T An example of non-decreasing solution for the KdV equation posed on a bounded interval
%J Comptes Rendus. Mathématique
%D 2014
%P 421-424
%V 352
%N 5
%I Elsevier
%R 10.1016/j.crma.2014.02.001
%G en
%F CRMATH_2014__352_5_421_0
Gleb Germanovitch Doronin; Fábio M. Natali. An example of non-decreasing solution for the KdV equation posed on a bounded interval. Comptes Rendus. Mathématique, Volume 352 (2014) no. 5, pp. 421-424. doi : 10.1016/j.crma.2014.02.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.02.001/

[1] J. Bona; H. Chen Periodic traveling-wave solutions of nonlinear dispersive evolution equations, Discrete Contin. Dyn. Syst., Volume 33 (2013), pp. 4841-4873

[2] J.L. Bona; S.M. Sun; B.-Y. Zhang A nonhomogeneous boundary-value problem for the Korteweg–de Vries equation posed on a finite domain, Commun. Partial Differ. Equ., Volume 28 (2003), pp. 1391-1436

[3] P.F. Byrd; M.D. Friedman Handbook of Elliptic Integrals for Engineers and Scientists, Springer, NY, 1971

[4] E. Cerpa Exact controllability of a nonlinear Korteweg–de Vries equation on a critical spatial domain, SIAM J. Control Optim., Volume 46 (2007), pp. 877-899

[5] E. Cerpa; E. Crépeau Boundary controllability for the nonlinear Korteweg–de Vries equation on any critical domain, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 26 (2009), pp. 457-475

[6] J.-M. Coron; E. Crépeau Exact boundary controllability of a nonlinear KdV equation with a critical length, J. Eur. Math. Soc., Volume 6 (2004), pp. 367-398

[7] A.V. Faminskii; N.A. Larkin Initial–boundary value problems for quasilinear dispersive equations posed on a bounded interval, Electron. J. Differ. Equ., Volume 1 (2010), pp. 1-20

[8] O. Goubet; J. Shen On the dual Petrov–Galerkin formulation of the KdV equation on a finite interval, Adv. Differ. Equ., Volume 12 (2007), pp. 221-239

[9] A. Jeffrey; T. Kakutani Weak nonlinear dispersive waves: a discussion centered around the Korteweg–de Vries equation, SIAM Rev., Volume 14 (1972), pp. 582-643

[10] G. Perla Menzala; C.F. Vasconcellos; E. Zuazua Stabilization of the Korteweg–de Vries equation with localized damping, Q. Appl. Math., Volume 60 (2002), pp. 111-129

[11] L. Rosier Exact boundary controllability for the Korteweg–de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., Volume 2 (1997), pp. 33-55

[12] B.-Y. Zhang Exact boundary controllability of the Korteweg–de Vries equation, SIAM J. Control Optim., Volume 37 (1999), pp. 543-565

Cité par Sources :

Commentaires - Politique